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The "multiplicative honeycomb" an

inclusive strategy for strengthening

mathematical reasoning and problem

solving skills.

El “panal multiplicativo” una estrategia inclusiva para el

fortalecimiento de habilidades de razonamiento y

resolución de problemas matemáticos

Germán Eduardo Bolaños Cerón

Christian Stalin Narvaez Zúñiga*

Leidy Dayana Karen Vallejo Morán*.

Juan Gabriel Castañeda Polanco*.

Abstract

The research project mentioned above was aimed at improving reasoning

and problem solving skills through the design of a ludic-pedagogical

strategy called "Multiplication Honeycomb", which contributed to the

improvement of the teaching-learning processes and the exercise of the

aforementioned skills. The research took place in the Municipal

Educational Institution Nuestra Señora de Guadalupe, located in the village

of Catambuco, Botanilla - Nariño, with fourth grade students, some of

* Degree in French English UDENAR - I.E.T.A San Carlos -

Colon Genova, Nariño, Corporación Universitaria

Iberoamericana, Wengger4@hotmail.com.

https://orcid.org/0000-0001-5332-2765

Degree in Basic Education - Emphasis on Natural Sciences

and Environmental Education UDENAR - I.E.T.A el Empalme

Ipiales Nariño, Corporación Universitaria Iberoamericana,

cristhian-898@hotmail.com, https://orcid.org/0000-0003-

0448-4519

Degree in Philosophy and Letters UDENAR - I.E.M Nuestra

Señora de Guadalupe - Catambuco, Nariño, Corporación

Universitaria Iberoamericana

dayanavallejomoran@hotmail.com. https://orcid.org/0000-

0002-0872-560X

Psychologist, Corporación Universitaria Iberoamericana

juan.castaneda@iberoamericana.edu. https://orcid.org/0000-

0001-7632-7526

Article

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whom present certain restrictions in the mathematical skills previously

mentioned, detected by means of the results of tests carried out in their

classes. The methods and types of research, were linked to quantitative

research, of quasi-experimental type and descriptive scope; understood

these as conjunctural forms, which comprise from dynamics of the context

and the need to implement strategies validated in applied exercises,

allowed to visualize realities and their superior results in mathematics from

the boys and girls of the Educational Institution. One of the fundamental

results that emerged was related not only to a better understanding and

mathematical skills, but also to the strengthening of coexistence and

socioemotional skills that arise from learning by doing and the

appropriation of the exercise of construction of this tool from the boys and

girls.

Key words: Teaching and learning strategies, multiplicative honeycomb,

mathematical skills, social-emotional skills and integration.

Resumen

El proyecto investigativo mencionado tuvo como, la mejora de las

habilidades de razonamiento y resolución de problemas mediante el diseño

de una estrategia lúdico – pedagógica denominada “Panal de

Multiplicación”, está contribuyó en el mejoramiento de los procesos de

enseñanza – aprendizaje y en la ejercitación de las habilidades

mencionadas. La investigación se dio lugar en la Institución Educativa

Municipal Nuestra Señora de Guadalupe, ubicada en el corregimiento de

Catambuco, vereda Botanilla – Nariño, con los estudiantes de grado cuarto,

algunos de los cuales presentan ciertas restricciones en las habilidades

matemáticas anteriormente señaladas, detectadas mediante los resultados

de pruebas realizadas en sus clases. Los métodos y tipos de investigación,

estuvieron ligados la investigación cuantitativa, de tipo cuasiexperimental

y alcance descriptivo; entendidas éstas como formas coyunturales, que

comprenden a partir de dinámicas propias del contexto y la necesidad de

implementar estrategias validadas en ejercicios aplicados, permitieron

visibilizar realidades y sus resultados superiores en matemáticas desde los

niños y las niñas de la Institución Educativa. Uno de los resultados

fundamentales, que emergieron, fue lo relacionados no sólo a una mejor

comprensión y habilidades matemáticas sino al fortalecimiento de la

convivencia y habilidades socioemocionales que surgen del aprender

haciendo y la apropiación del ejercicio de construcción de esta herramienta

desde los chicos y chicas.

Palabras clave: Estrategias de enseñanza aprendizaje, panal

multiplicativo, habilidades matemáticas, habilidades socioemocionales e

integración.

Introduction

The Guadalupe Municipal Educational Institution, main office is

located in the Municipality of Pasto, Catambuco, perimeter road

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sector, in the urban area of this sector. It is composed of 27

elementary school teachers at the main office and 812 elementary

school students of which 425 are male and 387 are female, these

students belong to low socioeconomic strata and with reduced

conditions in their social-emotional environments, which make it

difficult for them to interact proactively in the comprehensive

educational process.

Low performance in mathematics is a constant, presenting the loss

of the subject, without reaching the expected level. This problem

lasts until adulthood, having a considerable impact on the

development of everyday situations. The economic and social

repercussion involved with this phenomenon, is fundamental to

understand the processes implicit in the learning and execution of

mathematics Cragg & Gilmore (2014). In this sense, it is relevant to

mention that it not only has an impact on the issues mentioned by

the authors, but also generates frustration and therefore a reduction

in the motivation to enter higher education of boys and girls in the

IEM.

Within the comprehensive reading of the context through the non-

participant observation and what was referred by the teachers, it was

evidenced that, regarding the learning of the multiplication process,

it was necessary to strengthen the reasoning and problem-solving

skills, since most of the students presented difficulties in this aspect.

Taking into account the above and in order to generate a change in

the population, a strategy was designed to provide a solution to this

problem, strengthening the teaching-learning processes in the

exercise of reasoning and problem solving skills through didactic

material from a playful approach; for this purpose, the limitations

that the students had in the classroom in the development of these

skills were first identified and this led to the design of a playful

strategy through the multiplicative honeycomb.

As mentioned, through the construction of the multiplicative

honeycomb for the use and practice of mathematics in the fourth

grade of the IEM, it was possible to demonstrate the difficulties that

students have for a good performance in mathematics, as well as the

strategies to be used collectively that make possible, not only the

strengthening of this competence but also integral aspects of the

teaching-learning process, such as coexistence and personal skills in

each of the participants.

As a great result, it was identified that through research processes

such as the one proposed, it is possible to show skills that are not

very visible, due to the number of students inside a classroom, not

seen in the school day; proposing strategies that make it possible to

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demonstrate socioemotional and formative changes in children

where didactic strategies such as the multiplicative honeycomb are

used.

Materials and methods

In the teaching-learning process, it is worth highlighting the

importance of didactic resources within this process, as it could be

said that "without resources there is no learning". (Romero, 2016,

p.45). Didactic resource refers to any material that has been

elaborated with the intention of facilitating the teacher's function and

that of the student. It can be said that the didactic material is an

instrument that the teacher elaborates with the purpose of improving

the teaching-learning process in order to awaken the student's

interest in learning. Didactic material refers to those means and

resources that facilitate teaching and learning within an educational

context, stimulating the function of the senses to easily access the

acquisition of concepts, abilities, attitudes or skills.

The proposed methodology, after conducting an analysis of the boys

and girls, as well as a conceptual understanding of the definitions,

the researchers assumed quantitative research, understood as the set

of instruments that enable the objective definition of the

intervention, as well as the reliability of the resulting statements at

the end of the mathematical exercises proposed for the improvement

of mathematics in the fourth grade.

(Gamboa et al., 2019); von Feigenblatt (2020) The type of research

assumed is quasi-experimental, since it articulates previous

knowledge, team and participatory work, as well as the need for a

previous and subsequent diagnosis to understand the achievements

of the instruments applied before, during and after the

implementation of the multiplicative honeycomb.

The scope of the results found is descriptive; that is, the

understanding and teaching of mathematics to fourth grade students

made it possible to propose a comprehensive training strategy that

could possibly be implemented in other contexts, under similar

trends for the achievement of objectives that enable progress in the

learning of mathematics.

The population and participants were 10 students (6 boys and 4 girls)

of fourth grade of the IEM Nuestra Señora de Guadalupe, between

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the ages of 7 and 8 years, in the Botanilla - Nariño district; they are

located in a rural area, which as mentioned, are of strata between 1

and 2 in levels up to lower.

For the collection of information, a field diary was developed in

which the interactions between the students and the teacher, derived

from the process of development of mathematical skills, will be

recorded. Zurbano (2014) In addition, a semi-structured interview

will be used, where, depending on the development of the interview,

open questions of interest for the research will arise. This interview

will focus on the mathematics teacher and will also be applied to

other teachers working in the same grade.

It is expected that this research project has strengthened the skills of

reasoning and problem solving in multiplication for fourth grade

students of the I.E.M. Nuestra Señora de Guadalupe de Catambuco

- Nariño; with the report obtained, the results of the research were

contributed to the educational and scientific community through an

article published to date.

Results

The results presented below are based on the following population

sample.

Table 1. Description of the unit of analysis (sample).

EDUCATIONAL

INSTITUTION

CONTEXT

LEVEL

NO. OF

STUDENTS

I.E.M. Our lady

of Guadalupe

Rural

Primary

Source: this research. 2021

When reviewing the field diary, with respect to the difficulties that

arose with the children in fourth grade of the IEM, the following

stand out: repetition, temporary memory and lack of internalization

of concepts. A quantitative analysis of the evaluated guide revealed

the following:

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Figure 1. Difficulties presented in students.

Source: this research. 2021.

According to the diagnostic and/or perception interview conducted

with fourth grade teachers of the educational institution, the

following limitations in the problem-solving process were unified,

among others, poor reading, not looking for a strategy to solve the

case and not using didactic material to pose the problem.

Figure 2. Constraints analyzed by the teachers interviewed

N° students: 10 (100%)

Repetición Memoria temporal Carencia de interiorización

5 students

2 students

3 students

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Source: this research. 2021.

It was also identified regarding the mathematical reasoning process,

in fourth grade children: the lack of implementation of playful-

pedagogical strategies, the socialization of such strategies and

chaining of processes that manage to link the student in the process

of mathematical reasoning.

Figure 3. Constraints on mathematical reasoning analyzed by

teachers interviewed.

Source: this research. 2021.

The above allowed us to infer that there are clear, repeated and

transcendental problems in the teaching and learning process in

general and in mathematics, which are deep and serious and are

reproduced in the other grades, in educational institutions such as

0,5

1,5

2,5

3,5

4,5

Docente 1 Docente 2 Docente 3

Problem solving constraints

Low reading

Docente 1 Docente 2 Docente 3

Limitations of mathematical reasoning

Lack of playful implementation

Lack of socialization of strategies

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those mentioned, with low levels of dedication and academic

recognition in the municipality of Pasto. Now, according to the

interview conducted with fourth grade teachers from different

educational institutions in the municipality of Pasto, the following

limitations in the process of problem solving were unified:

Reading is fundamental for the complete formation of

students, and the acquisition of a reading habit is crucial for

their educational development. A major strategy for learning

and teaching is to encourage thinking and understanding the

text. Reading comprehension and the development of

intellectual skills is based on the interpretation of the text and

the contextualization of the problem (Frías, 2015, p. 30).

The context in which a mathematical problem is developed and its

correct interpretation is involuted by the use of technological tools,

based on diffuse and poorly argued literature extracts. The reading

habit in the population sample of this study is impaired, thus leading

to consequences in the approach of mathematical problems with

simple solutions based on a correct interpretation of the text.

Limited resolution strategies, according to Meneses and Peñaloza

(2019) "the resolution of each problem brings with it a great

discovery, the problem must challenge curiosity and induce to bring

inventive faculties into play".

Therefore, the learner must apply a learning procedure in a relevant

way, which implies possessing the basic skill that allows it, knowing

the procedure in question (procedural knowledge) and being able to

apply it appropriately when conditions make it advisable, according

to an objective (strategic use of the procedure).

With this scenario, the working group decided to adopt from its

construction, the implementation of the multiplicative honeycomb as

a training strategy that, as conceptually defined, enables the

spontaneous, participatory and playful learning qualification, from

and with the dynamics of the boys and girls present in the exercises.

This made it possible to identify, initially, the difficulties of the

students in the development of the exercise; one of them did not

respond correctly to the results of the first exercise, allowing the

researchers to determine the need to integrate the strategies of

socialization of the exercise to the levels of understanding of the

students.

In this sense and as mentioned by Mora, the possibility of employing

systematic methods in the evolution of mathematical reasoning is

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determined throughout the development of education. In early

childhood education, it favors the child's search for objects, which

could be translated into the use of numerical location methods

motivated by intuition. In Primary Education, playful procedures

would be applied, firstly, to numerical symbology, using

alphabetical sorting, and then to other reference works and guides

based on numerical or alphanumerical sorting (Mora 2014, p. 23).

After the construction of the multiplicative honeycomb, together

with its use in the mathematical exercises proposed in the research,

the students remained happy, they assumed the equations, numerical

results, mathematical concepts, multiplications through this tool;

assuming a proactive disposition, caring for their constructed

material, as well as understanding the alternatives in the teaching and

learning process derived from the multiplications through the

multiplicative honeycomb.

Figure 4. Post-test results with students

Source: this research. 2021

Discussion

The present research proposal is circumscribed within the

framework of the research line in inclusion, understood as the

educational process through which the university contributes from

the academic production, in the processes of inclusive

transformation in children of school age and that make possible the

integral formation in their educational environment.

The elements proposed for the implementation of the exercises, as

well as the analysis based on these comprehensions, made it possible

to identify the need to encourage the study of teaching and learning

strategies in all areas; however, for this research, it is necessary to

15,1

4,5 3,4 3,26

RESPUESTAS

CORRECTAS

RESPUESTAS

INCORRECTAS

PREGUNTAS

SIN

CALIFICACIÓN TOTAL

PREGUNTAS

POST - TEST

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prioritize the implementation of these strategies in the

comprehension of mathematics.

The elements to highlight then, are related to the teaching and

learning strategies of mathematics, the diagnosis or initial tests that

allowed us to observe the initial and/or problematic elements to

transform through the results after the application of the

multiplicative honeycomb, the resulting elements after the exercise,

the strengthened mathematical skills, as well as emerging elements

not identified in the formulation of the research, related to

motivation and coexistence that were enriched during the research

process with the students of the educational institution.

The theoretical reference was based on the following terms.

Teaching - Learning Process: this process is defined as "the

movement of the cognitive activity of students under the direction of

the teacher, towards the mastery of knowledge, skills, habits and the

formation of a scientific conception of the world." (Tobón, 2019,

78). In this process, the three elements of the educational cause are

combined (students, teachers and the object of knowledge), where

each element plays a different role depending on the methodological

moment of this academic training process, and where this

conjugation will result in the instruction to be able to solve and

understand various situations that arise in the life of both actors.

"Teaching - learning is a process, because such training takes place

in a systematic and progressive way, by ascending stages, each of

which is marked by quantitative changes that lead to qualitative

changes in students, in the cognitive, volitional, affective and

behavioral aspects". (Marchán-Carvajal & Sanmartí, 2015, p 90)

In other words, training on the need for the multiplicative

honeycomb in a participatory manner and with the manual

interaction of the students, allowed generating greater skills in the

understanding, reflection and subsequent participation of the

children in the exercises formulated for their development and

analysis.

The teaching and learning strategies should therefore integrate the

context, the skills of listening and execution of an exercise, the use

and participatory integration of all children in the exercises, enabling

the participation, inclusion and use of the resources present in their

environment, as well as the friendly interrelationship with their

peers.

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For Brito-Orta & Espinosa-Tanguma (2015) defines: "teaching aids

are all those components of the teaching-educational process that

serve as material support for teaching methods to enable the

achievement of the objectives set" p. 56. In this definition, teaching

aids are recognized as visual and auditory media as well as real

objects, textbooks, laboratories and all the material resources that

support the teacher's work.

It is vital then, that the means for meaningful teaching and learning

exercises are provided; these must consider the context, the

alternative difficulties for those who cannot use them or have

psychomotor difficulties, alternatives for the creation and

completion of this, as well as the significant needs where they will

see their final results reflected.

Regarding Mathematics Teaching-Learning: both students and

teachers have a determining influence on the success of the

mathematics teaching-learning process. Teacher and student are

responsible for the development and results of the didactic practice.

Both have to accept their strengths and weaknesses. The difficulties

in learning mathematics are largely related to the little action that

students have during the realization of mathematical activities. We

are in the presence, then, of a didactic problem, which can be solved

by means of a progressive conception of pedagogy.

Mathematical reasoning develops on the basis of executive

functions, where mathematical skills use a transversal correlational

design, in which working memory is implicit, involving verbal and

written calculation, as well as mathematical problems in the form of

statements. The numerical information received by humans from an

early age is directly affected by working memory and its gradual

development. This particular correlation involves the training of the

executive functions of the brain and its specific domain. Several

investigations coincide in a theoretical model that predicts the

relationships between executive function skills and the components

of mathematical knowledge Cragg & Gilmore (2014), These two

conceptual understandings, allow us to land the particular need for

this research, the use of the multiplicative honeycomb as a strategy

to strengthen mathematics in elementary school boys and girls, who

have limitations in the development of skills for numerical

reasoning.

The very fact of making this tool with their own hands allowed us to

identify the appropriation, motivation and need for natural

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understanding not only of the exercise, but also of mathematics by

making use of this tool created by themselves.

It could be argued that the learning of mathematics only takes place,

outside or inside school institutions, if students actually participate

in the development of mathematical concepts and ideas.

Mathematics is learned, just like other areas of scientific knowledge.

In general, the teaching of mathematics begins with a brief

motivating introduction, which enables students' interest and action,

according to their previous knowledge, personal intuition and

learning methods known to them as a result of their intra- and extra-

mathematical socialization process. (Abreu et al., 2018, 90)

It is significant for the researchers to understand the progress made

in the implementation of these activities in a participatory and

inclusive manner, understanding inclusion as "making the right to

education effective requires guaranteeing that all children and young

people have, first and foremost, access to education, not just any

education, but a quality education with equal opportunities. It is

precisely these three elements that define educational inclusion or

inclusive education". (Echeita and Duk, 2008, p. 98).

This was reflected in the characteristics that were highlighted in the

learning of the research process with the fourth grade children; in

this sense, integration exercises are promoted, shared activities,

personal reflection without discrimination among them, as well as

the recognition of diverse characteristics in people that make

possible a diverse learning process that does not exclude others.

With the results described, as well as what was demonstrated in the

tests after the exercises, it was possible to identify the transformation

in the understanding of mathematics with children in the fourth

grade; it is considered that the student achieves learning when he/she

can solve problems, some time after the education period and in a

wide variety of situations. With regard to teaching, according to the

constructivist approach there is no single way, the student should be

provided with different atmospheres, scenarios and/or environments

that allow him/her to build mathematical knowledge; it is necessary

to motivate him/her to explore ideas, make conjectures and then

justify them, so that through mathematical reasoning he/she can

argue his/her assertions and convince his/her peers and teachers. It

is also important to work on problem solving and to include real-

world applications.

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As described, the use of the multiplicative honeycomb, since its

creation spoiblit an additional motivation in the understanding of

mathematics; according to Maslow, referred to in the Manual of

Psychology and Educational Development volume 1, 2008 by

Herrería Valero: "children learn by playing, without worrying about

the nature of learning" (p.53). Ángel (2006) states: "the child learns

from action on objects". This refers to the fact that the child learns

by maneuvering objects in order to discover and register his

environment. It is not the same for him to discover as for the teacher

to impose on him what he should do and how he should do it; the

educator should be a guide to improve his expectations of the topics

presented and make the teaching-learning process more interactive.

In this sense and in defense of strategies such as the multiplicative

honeycomb, Sánchez Macías, a graduate in Mathematical Sciences,

stresses his uneasiness about the design of materials that help to

provide an adequate follow-up of teaching-learning of mathematics

from early ages until their university life, which implies that teachers

of all areas and mainly of the first grades, should have a similar sense

of concern with the purpose of creating and implementing new ways

that lead to a meaningful and attractive learning in students through

playful-pedagogical practices.

With the observational data practiced with students, there are good

inventive faculties, however, they are mostly not cohesive with

mathematical skills.

According to the data collected in the observation of the population

sample, as well as in the interviews conducted with the different

teachers, it is observed that there are not enough didactic alternatives

to support mathematical concepts, especially those centered on

multiplication.

The second criterion compiled is the consideration of procedures as

contents; that is, as culturally organized forms of knowledge that

have epistemological relations between them. In this sense, in the

teaching of two synthesis procedures, elaborating schemes and

constructing concept maps, we could put the former before the latter,

which is based on their requisite relations. The construction of maps

involves great difficulty if the student has not previously been taught

to group a set of ideas in a schematic way.

The positive effects of the observation and analysis led to the

creation of the "Multiplicative honeycomb", not only as a didactic

strategy but also as the central object of study of this proposal.

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